Question: Simplify and expand the following expression: $ \dfrac{2}{a - 10}- \dfrac{2}{3a + 24}- \dfrac{a}{a^2 - 2a - 80} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the second term: $ \dfrac{2}{3a + 24} = \dfrac{2}{3(a + 8)}$ We can factor the quadratic in the third term: $ \dfrac{a}{a^2 - 2a - 80} = \dfrac{a}{(a - 10)(a + 8)}$ Now we have: $ \dfrac{2}{a - 10}- \dfrac{2}{3(a + 8)}- \dfrac{a}{(a - 10)(a + 8)} $ The least common multiple of the denominators is: $ (a - 10)(a + 8)$ In order to get the first term over $(a - 10)(a + 8)$ , multiply by $\dfrac{3(a + 8)}{3(a + 8)}$ $ \dfrac{2}{a - 10} \times \dfrac{3(a + 8)}{3(a + 8)} = \dfrac{6(a + 8)}{(a - 10)(a + 8)} $ In order to get the second term over $(a - 10)(a + 8)$ , multiply by $\dfrac{a - 10}{a - 10}$ $ \dfrac{2}{3(a + 8)} \times \dfrac{a - 10}{a - 10} = \dfrac{2(a - 10)}{(a - 10)(a + 8)} $ In order to get the third term over $(a - 10)(a + 8)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{a}{(a - 10)(a + 8)} \times \dfrac{3}{3} = \dfrac{3a}{(a - 10)(a + 8)} $ Now we have: $ \dfrac{6(a + 8)}{(a - 10)(a + 8)} - \dfrac{2(a - 10)}{(a - 10)(a + 8)} - \dfrac{3a}{(a - 10)(a + 8)} $ $ = \dfrac{ 6(a + 8) - 2(a - 10) - 3a} {(a - 10)(a + 8)} $ Expand: $ = \dfrac{6a + 48 - 2a + 20 - 3a}{3a^2 - 6a - 240} $ $ = \dfrac{a + 68}{3a^2 - 6a - 240}$